## 1. Introduction

[2] Transmissivity (*T*) and storage coefficient (*S*) are two important properties that control groundwater flow in aquifers and are of practical importance for water resources development and management as well as protection and remediation of groundwater. After *Yeh and Liu* [2000] and *Liu et al*. [2002] sequential pumping tests, multiwell interference tests, or hydraulic tomography (HT) have been the subject of active theoretical, laboratory, and recently field research to characterize the spatial distributions of hydraulic parameters at a higher level of detail than traditional methods [e.g., *Gottlieb and Dietrich*, 1995; *Yeh and Liu*, 2000; *Liu et al*., 2002; *Bohling et al*., 2002, 2007; *Brauchler et al*., 2003, 2011; *Zhu and Yeh*, 2005, 2006; *Liu et al*., 2007; *Ni and Yeh*, 2008; *Straface et al*., 2007; *Xiang et al*., 2009; *Illman et al*., 2007-2011; *Fienen et al*., 2008; *Kuhlman et al*., 2008; *Castagna and Bellin*, 2009; *Berg and Illman*, 2011; *Cardiff et al*., 2009, 2012; *Cardiff and Barrash*, 2011; *Huang et al*., 2011; *Li et al*., 2005, 2008; *Liu and Kitanidis*, 2011; *Yin and Illman*, 2009]. Note that *Cardiff and Barrash* (2011) provided a summary of all peer-reviewed HT studies (1D/2D/3D).

[3] These studies showed that transient HT can identify not only the pattern of the heterogeneous hydraulic conductivity (*K*) or transmissivity (*T*) field, but also the variation of specific storage (*S _{s}*) or the storage coefficient (

*S*) [see

*Zhu and Yeh*, 2005, 2006;

*Liu et al*., 2007;

*Xiang et al*., 2009, in particular]. More importantly, they have demonstrated that the hydraulic property fields estimated by HT can yield much better predictions of flow and solute transport processes than other conventional characterization approaches [

*Illman et al*., 2011].

*Berg and Illman*[2011], in particular, substantiated the robustness of HT for a highly heterogeneous geological medium with a variance of log hydraulic conductivity of 5.4 and a vertical correlation scale of 0.15 m.

[4] Nonetheless, many practical issues remain, including (1) the design of spatial sampling and pumping locations, (2) the duration and magnitude of the pumping rate during an HT survey, and (3) the frequency of temporal sampling. Specifically, is it necessary to conduct the pumping test to reach a steady-state? How many time/drawdown data in a well hydrograph should be used to obtain good estimates of *T* and *S* field while minimizing computational effort for the HT analysis? While *Yeh and Liu* [2000] conducted a preliminary investigation of the spatial sampling issues, the temporal sampling issues are opened for investigation.

[5] The sampling time requirements vary with the method of interpretation. For instance, *Vasco et al*. [2000], *Vasco and Karasaki* [2006], *Brauchler et al*. [2003, 2011], and *He et al*. [2006] developed methods based on travel time and amplitude of drawdown-time curves at different locations from the pumping tests conducted in tomographic format to estimate hydraulic properties. Similar to the travel time approaches, *Li et al*. [2005] and *Zhu and Yeh* [2006] developed temporal moment approaches, which are based on multidimensional flow model. These approaches either implicitly or explicitly require a complete well hydrograph to determine the travel time and the amplitude (i.e., the temporal moments).

[6] On the other hand, *Yeh and Liu* [2000], *Fienen et al*. [2008], and *Liu and Kitanidis* [2011] developed HT interpretation methods based on steady-state flow models which require only steady-state heads or drawdowns. *Zhu and Yeh* [2005] reported that heads are highly correlated in time during HT experiments, As a result, *Zhu and Yeh* [2005], *Liu et al*. [2007], *Straface et al*. [2007], *Xiang et al*. [2009], *Illman et al*. [2009], *Castagna et al*. [2011], *Berg and Illman* [2011], *Cardiff and Barrash* [2011], and *Cardiff et al*. [2012] analyzed transient HT data using only a small numbers (four or five) of selected heads or drawdowns of the drawdown-time well hydrograph. *Bohling et al*. [2002], however, used 51 drawdown-time data points of each well hydrograph of 42 sampling ports of 14 pumping tests (a total of 29,988 drawdown data points) for transient inversion for hydraulic conductivity field only, assuming the *S* field was known exactly. Because of the large amount of head data used, the amount of computational resource required was overwhelming. Consequently, they applied the concept of steady shape to their analysis of HT data for the *K* field. The term “steady shape” is used to designate conditions in an unsteady-state flow regime in which drawdown continues to change with time but the hydraulic gradient remains constant. Specifically, they suggested that the head differences between different observation points (hydraulic gradient) can be analyzed using steady-shape flow model during the steady-shape regime even though the drawdown remains transient. *Hu et al*. [2011] similarly used the steady-shape concept and travel time approach to analyze the HT survey. *Bohling* [2009] further claimed that the steady-shape approach reduces the influence of uncertainty in boundary conditions when compared with the approach that uses drawdowns in a steady-state flow model. *Bohling et al*. [2002], *Bohling* [2009], and *Bohling and Butler* [2010] also advocated that neither steady-shape nor transient approach for HT analysis was capable of revealing hydraulic conductivity variations outside of the region encompassed by the pumping and observation wells.

[7] This paper investigates the time of drawdown measurements of well hydrographs that can maximize the resolutions of *T* and *S* estimates in the analysis of a HT survey. We employed a first-order cross-correlation analysis to investigate the temporal and spatial evolutions of cross-correlation between the head at an observation well and *T* and *S* in homogeneous and heterogeneous aquifers during a pumping test. Based on the analysis, we explored the head information content at different time periods during a pumping test on spatial distribution of *T* and *S*, then proposed a temporal sampling strategy for HT analysis, and at last tested it using numerical examples with and without prior knowledge of the boundary conditions. At the end, the cross-correlation analysis is used to explain the robustness of HT on *T* estimation, even beyond well field, and also to explain the limitation of HT on *S* estimation.